Essentially the same difference between vectors and arrays. A vector is an element of a vector space. A vector can be represented as a coordinate array in a bas

Essentially the same difference between vectors and arrays. A vector is an element of a vector space. A vector can be represented as a coordinate array in a basis, but it is not the coord array. Consider the vector 1 + x + x^2 you might use the canonical basis for polynomials {1, x, xx, xxx, ...} and then the coordinate vector is [1 1 1]. The coordinate and the basis represent the polynomial. But the polynomial isn't a list of numbers. A matrix is the coefficient list of a linear transformation, but it is not the transformation. The derivative operator is a linear transformation. It's representation as a matrix is just coordinates. But the operator is clearly more than that. A tensor can be written down and you can compute with the components, but the tensor itself is the basis independent transformation. Not a list of components.